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United States Patent |
6,239,878
|
Goldberg
|
May 29, 2001
|
Fourier-transform and global contrast interferometer alignment methods
Abstract
Interferometric methods are presented to facilitate alignment of
image-plane components within an interferometer and for the magnified
viewing of interferometer masks in situ. Fourier-transforms are performed
on intensity patterns that are detected with the interferometer and are
used to calculate pseudo-images of the electric field in the image plane
of the test optic where the critical alignment of various components is
being performed. Fine alignment is aided by the introduction and
optimization of a global contrast parameter that is easily calculated from
the Fourier-transform.
Inventors:
|
Goldberg; Kenneth A. (Berkeley, CA)
|
Assignee:
|
The Regents of the University of California (Oakland, CA)
|
Appl. No.:
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409646 |
Filed:
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October 1, 1999 |
Current U.S. Class: |
356/520; 356/508; 356/521 |
Intern'l Class: |
G01B 009/02 |
Field of Search: |
356/508,511,520,521
|
References Cited
U.S. Patent Documents
5959730 | Sep., 1999 | Wang et al.
| |
Other References
Goldberg, Kenneth A., "Extreme Ultraviolet Interferometry", Ernest Orlando
Lawrence Berkeley National Laboratory, pp. 1-275, Dec. 1997, Ph.D. Thesis.
Takaki, Y. et al., "Hybrid holograhic microscopy free of conjugate and
zero-order images", Applied Optics, vol. 38, No. 23, 10 Aug. 1999, pp.
4990-4996.
|
Primary Examiner: Kim; Robert
Attorney, Agent or Firm: O'Banion; John P.
Goverment Interests
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
The United States Government has rights in this invention pursuant to
Contract No. DE-AC03-76SF00098 between the United States Department of
Energy and the University of California.
Claims
What is claimed is:
1. An alignment method wherein an interferometer is used to reduce
alignment error between an optical system and an image plane, comprising
the steps of:
(a) illuminating the optical system to produce resultant beams which strike
the image plane of an interferometer mask;
(b) selectively masking the resultant beams;
(c) recording beam intensity images for the resultant beams that are
transmitted through the selective mask; and
(d) calculating logarithmically-scaled magnitudes for the Fast
Fourier-transforms which are computed from the beam intensity images to
thereby produce alignment information.
2. A method as recited in claim 1, further comprising the steps of moving
at least one optical component to alter the position wherein at least one
resultant beam strikes the image plane mask, said move being made
substantially according to said alignment error information whereby an
effective reduction in alignment error is thereby produced.
3. A method as recited in claim 1, wherein said interferometer comprises a
phase-shifting point-diffraction interferometer.
4. A method as recited in claim 1, wherein said step of recording beam
intensity images comprises the steps of:
(a) impinging wavefronts from at least one beam onto the surface of a CCD
imager; and
(b) periodically capturing signals generated by the CCD.
5. A method as recited in claim 1, wherein the resultant beams are
selectively masked by passing a test beam through a test window in the
interferometer mask and additionally passing at least one reference beam
through at least one reference pinhole in said interferometer mask.
6. A method as recited in claim 1, wherein said step of calculating
logarithmically-scaled magnitudes for the Fast Fourier-transforms is
performed in near real time.
7. A method as recited in claim 1, wherein said step of illuminating the
optical system is performed by transmitting radiation from a source
through an object pinhole followed by spatial separation of the beam by a
beamsplitter.
8. A method as recited in claim 1, wherein the alignment information is
displayed in the form of a pseudo-image computed from the
logarithmically-scaled magnitudes for the Fast Fourier-transforms of the
intensity impinging on the imager, which is used in said moving of at
least one optical component according to said alignment information.
9. An alignment method wherein a interferometer is used to reduce alignment
error between a mask and an optical system's image plane, comprising the
steps of:
(a) illuminating the optical system to produce resultant beams which strike
the image plane of an interferometer mask;
(b) selectively masking the resultant beams;
(c) recording beam intensity images for the resultant beams that are
transmitted through the mask;
(d) capturing of at least one beam within the image plane mask aperture of
the interferometer;
(e) calculating the global fringe contrast parameter from Fast
Fourier-transforms which are computed from the beam intensity images to
thereby produce fine alignment information; and
(f) moving at least one optical component to alter the position where the
resultant beam intercepts the mask, said move being made substantially
according to said alignment information.
10. An inspection method wherein an interferometer is used for viewing an
appropriately-scaled pseudo-image of transmissive features of an
interferometer mask, comprising the steps of:
(a) illuminating the optical system to produce a resultant reference beam
which strikes the image plane of an interferometer mask;
(b) selectively masking the reference beam through a reference pinhole;
(c) recording beam intensity images for the resultant beam which is
transmitted through the selective mask; and
(d) calculating logarithmically-scaled magnitudes for the Fast
Fourier-transforms which are computed from the beam intensity images to
thereby produce an image in which transmissive features of the mask are
seen magnified whereby the method can be used for microscopic viewing of a
mask or optics used within the interferometer.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
Not Applicable
REFERENCE TO A MICROFICHE APPENDIX
Not Applicable
INCORPORATION BY REFERENCE
The following publications which are referred to in this specification
using numbers inside square brackets (e.g., [1]) are incorporated herein
by reference:
[1] Medecki, H., E. Tejnil, K. A. Goldberg, and J. Bokor, "Phase-shifting
point diffraction interferometer," Optics Letters, 21 (19), 1526-28
(1996).
[2] Tejnil, E., K. A. Goldberg, S. H. Lee, H. Medecki, P. J. Batson, P. E.
Denham, A. A. MacDowell, J. Bokor, and D. T. Attwood, "At-wavelength
interferometry for EUV lithography," Journal of Vacuum Science &
Technology B, Nov.-Dec. 1997, 15 (6), pp. 2455-61.
[3] Williamson, D. M., "The elusive diffraction limit," in OSA Proceedings
on Extreme Ultraviolet Lithography, Vol. 23, F. Zernike and D. T. Attwood,
Eds., Optical Society of America, Washington, D.C., 1994, pp. 68-76.
[4] Naulleau, P., K. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson,
D. Attwood, J. Bokor, "Characterization of the accuracy of EUV
phase-shifting point diffraction interferometry," Proc. SPIE, 3331, Santa
Clara, Calif., February, 1998, pp. 114-23.
[5] Naulleau, P., and K. A. Goldberg, "Dual-domain point diffraction
interferometer," submitted to Applied Optics, Sep. 1, 1998.
[6] Goodman, J. W., Introduction to Fourier Optics, Second ed.,
McGraw-Hill, New York, 1988.
[7] Goldberg, K., EUV Interferometry, doctoral dissertation, Physics
Department, University of California, Berkeley, 1997.
[8] Takeda, M., H. Ina, and S. Kobayashi, "Fourier-transform method of
fringe-pattern analysis for computer-based topography and interferometry,"
J. Opt. Soc. Am., 72 (1), 156-60 (1981).
[9] Nugent, K. A., "Interferogram analysis using an accurate fully
automatic algorithm," Applied Optics, 24 (18), 3101-5 (1985).
BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention pertains generally to testing an optical system with an
interferometer, and more specifically to methods by which a coherently
illuminated optical system can be aligned within an interferometer being
used for measuring or inspecting that optical system.
2. Description of the Background Art
Interferometers are often used for taking optical measurements on an
optical system. The process of photo-lithography, for example, employs a
variety of such optical systems which must be checked for errors and
aberrations. In order to accurately perform these tests, it is critical
that the components within the optical system be aligned using an
interferometer. One form of interferometer is a phase-shifting
point-diffraction interferometer (PS/PDI). The PS/PDI [1] generates a
spherical reference beam by pinhole diffraction in the image plane of an
optical system under test. A PS/PDI is shown in FIG. 1 being used with an
optic under test and a CCD detector. A monochromatic beam is diffracted by
an entrance pinhole spatial filter and then passed through a coarse
grating beamsplitter placed before the image plane on the object-side (or
alternately, the image-side) of the optic under test. The beamsplitter
generates multiple focused beams that are spatially separated in the image
plane. One of the beams from the test optic is allowed to pass through a
large window called the test window, within a patterned screen that is
herein referred to as a "mask".
The mask used may be either a transmissive mask or a reflective mask. In a
transmissive mask the selected area of the test window contains
transparent features, such as alignment marks, or may contain a fully
transparent window. In a reflective mask, which is often used for EUV
radiation, similar features or windows are selectively reflective. Use of
the transmissive form of mask element is generally described and depicted
herein, as it is easier to visualize and to understand; although either
form of mask element may be used within the inventive method. The mask is
located in the image plane and the beam passing through the test window is
referred to as the test beam.
Any beam so "chosen" by the selective masking contains nearly identical
aberration information about the optical system. A second beam from the
test optic can be brought to focus on a reference pinhole smaller than the
diffraction-limited resolution of the test optic, where it is spatially
filtered to become a spherical reference beam covering the numerical
aperture of measurement. A controllable phase-shift between the test and
reference beams is achieved by a simple lateral translation of the grating
beamsplitter. The test and reference beams propagate from the image plane
to a detector where the interference pattern is recorded. The detector is
positioned to capture the numerical aperture of measurement, and may be
used with or without re-imaging optics.
The PS/PDI has been successfully used in the measurement of
multilayer-coated, all-reflective extreme ultraviolet (EUV) optical
systems, operating near 13-nm wavelength [2], where the fabrication
tolerances are in the sub-nanometer regime [3]. Using pinholes on the
order of 100-nm diameter, two-mirror optical systems with numerical
aperture (NA) of 0.06-0.09 and system wavefront aberration magnitudes on
the order of 1-nm rms have been measured. Two-pinhole null tests have
recently verified the high accuracy (0.004 waves, or 0.054 nm rms within
0.082 NA) that is attainable with the EUV PS/PDI [4].
During the alignment process, the test window of the mask is normally
positioned to be centered on the test beam focus when the reference beam
is properly captured and centered on the reference pinhole. The test
window width in the direction of beam separation should be less than the
beam separation distance to minimize the undesirable overlap of the
reference beam through the window. In the EUV application, with a typical
beam separation of 4.5 .mu.m (27 times .lambda./NA), the window widths are
chosen to be 4.5 .mu.m or less. An additional constraint may be imposed to
achieve the complete separation of the orders in the Fourier domain of the
recorded intensity image; here the window width must be limited to
two-thirds of the beam-separation distance. [5]
Considering the small pinholes used in the measurement of high-resolution
optical systems, alignment is the most challenging aspect of using an
interferometer such as the PS/PDI. This fact is compounded in
short-wavelength applications where the interferometer exists inside of a
vacuum chamber and may be incompatible with other optical alignment
strategies. While the test beam is typically easy to align through the
large image-plane window, the reference beam should be positioned onto the
reference pinhole to within a fraction of the focal spot diameter. The
small size required of this pinhole attenuates the reference beam and
narrows the "capture range" over which interference fringes are visible.
Until the reference pinhole is within the focus of the reference beam,
only subtle clues are available to guide the alignment. During fine
alignment, once the beam has been captured, the intensity of the test beam
remains fixed, and proper positioning can be judged by assessing the point
of peak fringe contrast.
BRIEF SUMMARY OF THE INVENTION
The present invention pertains to optical alignment and viewing methods
that are based on the use of Fast Fourier-Transforms (FFT) performed on
detected images for use with an interferometer for measuring and testing
high resolution optical systems. As an alignment tool, the methods provide
for rapid alignment wherein the need of high accuracy equipment can in
some instances be eliminated. As a pseudo-microscope, the methods provide
a simple way in which to perform a magnified inspection of the mask used
within the interferometer. The inventive methods are described emphasizing
the qualitative description, and several simplifications are made to
illustrate the behavior of this method in a number of common
configurations.
An object of the invention is to provide a method that simplifies the
alignment of a test optic within an interferometer with an optical
detector.
Another object of the invention is to provide a method in which the
reference beam may be quickly aligned with the reference pinhole prior to
beam capture within the pinhole.
Another object of the invention is to provide a faster and more accurate
method of fine alignment, wherein a reference beam already captured within
a reference pinhole is precisely centered.
Another object of the invention is to provide a method whereby defects in
the mask used within an interferometer can be magnified for inspection or
measurement.
Further objects and advantages of the invention will be brought out in the
following portions of the specification, wherein the detailed description
is for the purpose of fully disclosing preferred embodiments of the
invention without placing limitations thereon.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention will be more fully understood by reference to the following
drawings which are for illustrative purposes only:
FIG. 1 is a schematic diagram of a PS/PDI interferometer with a test optic
upon which the Fourier-transform methods of the present invention are
used.
FIG. 2 a diagrammatically depicts transmissive mask pattern and
pseudo-images of intermediate and final alignment achieved using the
Fourier-transform methods of the present invention.
FIG. 3 is a schematic diagram of a transmissive mask in the PS/PDI shown
with test and reference beams interacting with the test window and
reference pinhole.
FIG. 4 diagrammatically depicts images of detected intensity from the test
optic with corresponding pseudo-images achieved using the
Fourier-transform methods of the present invention.
FIG. 5 diagrammatically depicts a set of corresponding images and
pseudo-images in which the beamsplitter of FIG. 4 has been rotated
90.degree..
FIG. 6 diagrammatically compares pseudo-images achieved using the
Fourier-transform methods of the present invention and images of the same
pair of masks created by a scanning electron microscope.
DETAILED DESCRIPTION OF THE INVENTION
Referring more specifically to the drawings for illustrative purposes, the
Fourier-transform and global contrast interferometer alignment method of
the present invention will be described with reference to FIG. 1 through
FIG. 6. It will be appreciated that the apparatus may vary as to
configuration and as to details of the parts and that the method may vary
as to the specific steps and their sequence without departing from the
basic concepts as disclosed herein.
1. Test System Overview
In FIG. 1 a PS/PDI form of interferometer 10 is shown upon which the
Fourier-transform and global contrast interferometer alignment method of
alignment may be performed. A narrow band, or monochromatic radiation
source 12 produces a narrow band of radiation frequencies 14 which are
passed through a pinhole diffractor 16, which produces a spherical beam 18
by diffraction. The spherical beam is passed through a coarse-grating
beamsplitter 20 that produces multiple focused beams comprising a source
test beam 22 onto the test optic 24. The source test beam is transmitted
through the test optic 24 which can produce a resultant test beam 26, and
a resultant reference beam 28. Both beams converge on a PS/PDI mask 30.
The resultant test beam 26, from the test optic 24 is allowed to pass
through a large window 32 in an otherwise opaque membrane comprising the
PS/PDI mask 30, located in the image plane, while the reference beam is
brought to focus on a reference pinhole 34 on the PS/PDI mask. The
resultant far-field test beam 38 and far-field reference beam then impinge
on a CCD detector 40 where the interference patterns are recorded. The
reference pinhole 34 is smaller than the diffraction-limited resolution of
the test optic. Transmission through the pinhole spatially filters the
reference beam, producing a spherical reference beam that covers the
numerical aperture of measurement. A controllable phase-shift between the
test and reference beams is achieved by performing a simple lateral
translation of the grating beamsplitter. Additional re-imaging optics may
be included prior to the detector. The CCD detector 40 is positioned to
capture the numerical aperture of measurement.
2. Theory
This section describes the mathematical basis of the alignment method and
illustrates, using several examples, the observable behavior of the system
during alignment.
Owing to its position in the far-field of the image plane, images recorded
by the detector approximate the intensity, or square-modulus, of the
Fourier-transform of the field transmitted through the PS/PDI image-plane
mask. This transmitted field may be viewed simply as the product of the
incident field from the test optic, and the transmission function of the
mask. With single-beam coherent illumination, in the absence of the
grating beamsplitter, this incident field is the point-spread-function of
the test optic. The transmission function of the mask is determined by the
pattern of opaque and transparent regions and by the lateral position of
the mask in the image plane.
In a single exposure, phase information is temporarily lost and the
detector measures only the field intensity. The Auto-correlation Theorem
[6] can be applied to the measured intensity to recover information about
the field transmitted through the image-plane mask. The transmitted
image-plane field is labeled as a (.rho.), where .rho. is a spatial
coordinate vector in the image plane. In the Fraunhofer approximation [6]
for the propagation of light, from the image plane to the detector, the
measured intensity is related to the Fourier-transform of a (.rho.)
evaluated at angular frequencies f=r/.lambda.z where r is a spatial
coordinate in the detector plane, and z is the distance to the detector
plane. The detected field is .vertline.A(r).vertline..sup.2, where A and a
form a Fourier-transform pair. When .vertline.A(r).vertline..sup.2 is
known, the following relations in two-dimensions hold
.Fourier.{.vertline.A(.rho.).vertline..sup.2
}=.Fourier.{A(.rho.)A*(.rho.)}=.intg.a(.rho.')a*(.rho.-.rho.')d.rho.'
(1a)
.tbd.a(.rho.).degree.a*(-.rho.). (1b)
where .Fourier. signifies the Fourier-transform, and .degree. denotes the
convolution operator.
2.1 Test Beam Alone
The field transmitted through the mask a(.rho.) is the product of the
incident field and the mask transmission function m(.rho.). Incident on
the mask is a focused beam, plus a lower level of scattered light and
flare. For demonstration, we may approximate the focused beam as a
delta-function, and this light near the focus as having constant magnitude
c, much less than unit intensity. m(.rho.) includes the large window for
the test beam and any light transmitted through the reference pinhole(s).
We approximate a(.rho.) as:
a(.rho.).apprxeq.[.delta.(.rho.)+c]m(.rho.-.rho..sub.0). (2)
where .rho..sub.0 is the lateral displacement of the mask relative to an
arbitrary coordinate origin. When the focus of the test beam is
unobstructed by the mask window, a(.rho.) becomes
a(.rho.).apprxeq..delta.(.rho.)+cm(.rho.-.rho..sub.0); (3)
otherwise, the mask blocks the focused portion of the test beam and
a(.rho.) contains only cm(.rho.-.rho..sub.0). Applying the approximation
of Eq. (3) to Eq. (1), we have
.Fourier.{.vertline.A(r).sup.
2.vertline.}.apprxeq.[.delta.(.rho.)+cm(.rho.-.rho..sub.
0)].degree.[.delta.*(-.rho.)+cm*(.rho..sub.0 -.rho.)] (4a)
.apprxeq..delta.(.rho.)+[cm(.rho.-.rho..sub.0)+cm*(.rho..sub.0
-.rho.)]+c.sup.2 m(.rho.).degree.m*(-.rho.). (4b)
In this way, the Fourier-transform of the measured intensity is separable
into three components: a delta-function peak at the origin; the low-level
mask transmission function and its polar-symmetric complex conjugate
folded about the point .rho..sub.0 and symmetric about the origin; and the
auto-correlation of the mask transmission function.
With magnitude proportional to c.sup.2, the auto-correlation of m forms a
very-low-level background about the central frequencies. The shape and
extent of the auto-correlation of m depends on the shape of the window.
The auto-correlation reaches a maximum at the central-frequency and
decreases to zero at the position that corresponds, respectively, to the
width of the window in each direction.
A PS/PDI mask transmission function 50 is shown in FIG. 2 with a large test
window 52 and a vertically displaced reference pinhole 54 and a
horizontally displaced reference pinhole 56. To the right of the mask
transmission function is shown a simple illustration 60 of the
auto-correlation described by Eq. (4). The pseudo-image illustration 60
represents detection as the resultant test beam from the test optic is
aligned to pass through the upper-left corner of the test window 52. The
pseudo-image 60 shows the magnitude of the Fourier-transform of the
detected field, representing the auto-correlation of the image-plane
transmitted field.
As the beam, or the mask, is translated laterally the change in .rho..sub.0
shifts only the positions of m and m* in the spatial-frequency domain;
which is the basis of the Fourier-transform alignment method. By observing
the motion of m and m* in the spatial-frequency domain (i.e., in the
Fourier-transform of the detected intensity), proper alignment of the test
beam in the mask window can be achieved. Proper alignment 70 is
illustrated in FIG. 2 with the vertical alignment of the three window
images. Owing to the speed of modern computers, this alignment can be
performed in near real-time.
This method can also be used as a coarse test of focus, or alternately the
longitudinal alignment of the interferometer. When the mask is displaced
from the image plane, the apparent size of the focal spot increases, and
the delta-function approximation of Eq. (3) must be replaced by a function
having a finite width. In practice, this results in a noticeable blurring
of the sharp window-edge features in the Fourier-transform of the measured
intensity. In the presence of moderate defocus, longitudinal alignment can
be performed while the "sharpness" of these features is assessed.
2.2 Test Beam with Reference Beams
When a beamsplitter is used in the PS/PDI, the field from the test optic
consists of a pair or a series of displaced copies of its
point-spread-function. The description of a(.rho.) above can be modified
to include the other orders as follows. With a beam separation vector s
between adjacent orders, and a series of displaced test beam copies of
peak magnitudes given by the coefficients {b.sub.j }, a(.rho.) becomes
##EQU1##
If only one beam is transmitted through the mask window, then the situation
will appear identical to Eqs. (3) and (4) with an increased magnitude of c
representing the combined flare from multiple beams.
When the PS/PDI is properly aligned for interferometry, the test beam
passes through the test window as shown by the mask and impinging beam
combination 80 of FIG. 3. The mask 82 is shown with a test beam 84 and a
pair of reference beams 86, 88 at 1.sup.st and --1.sup.st order. The test
beam 84 passes through the test window 90 of the mask 82, while a
reference beam 86 (1.sup.st order) is transmitted through the reference
pinhole 92 which is located at a position displaced by s, from the center
focal point of the test window. The other reference beam 88 is blocked by
the mask. A resultant test beam 94 and reference beam 96, emerge on the
far-field with interference 98. With the addition of the reference beam,
interference fringes become visible in the detected intensity. This second
beam in the image plane field causes the Fourier-transform of the detected
intensity to take a different appearance. In this situation the reference
beam in the image plane may be approximated as an additional displaced
delta function of relative peak magnitude d.
a(.rho.).apprxeq.[.delta.(.rho.)+d.delta.(.rho.-s)+c]m(.rho.-.rho..sub.0)
(6a)
The small size and spatial filtering of the reference pinhole reduce the
reference beam magnitude to a new magnitude d' upon transmission. Now the
approximation for the transmitted field in the image plane is
a(.rho.).apprxeq..delta.(.rho.)+d'.delta.(.rho.-s)+cm(.rho.-.rho..sub.0).
(6b)
Relative to Eq. (4) in the single beam case, the auto-correlation of
a(.rho.) here contains several additional terms.
.Fourier.{.vertline.A(r).vertline..sup.2
}.apprxeq.[.delta.(.rho.)+d'.delta.(.rho.-s)+d'.delta.*(s-.rho.)]+[cm(.rho
.-.rho..sub.0)+cm*(.rho..sub.0 -.rho.)+cd'm(.rho.-.rho..sub.0
-s)+cd'm*(s+.rho..sub.0 -.rho.)]+c.sup.2 m(.rho.).degree.m*(-.rho.). (7)
There are now three narrow peaks in the Fourier-transform: one at the
origin, and two displaced by the beam separation vector .+-.s,
representing the reference beam and its complex conjugate. In addition to
the two overlapped, polar-opposite mask patterns at the origin, m(.rho.)
is repeated at s and -s. This situation of proper alignment 70 is
illustrated in FIG. 2 with the vertical alignment of the three window
images. There is a relative magnitude difference of d' between the central
and the displaced patterns due to the intensity difference of the test and
reference beams. The background auto-correlation of m(.rho.) is still
present in the same form as before.
Since these additional components of the Fourier-transform appear only when
the reference beam is aligned to pass through the reference pinhole, their
presence indicates that the PS/PDI is aligned for interferometric
measurements. An additional fine alignment merit-function, called Global
Image Contrast, is described in Section 4.
2.3 Reference Beam Alone
A final case for consideration is that of the reference beam alone. When a
single beam passes through the reference pinhole and only scattered light
is transmitted through the adjacent window, the situation can be
approximated as
a(.rho.).apprxeq.[d.delta.(.rho.)+c]m(.rho.-s) (8a)
.apprxeq.d'.delta.(.rho.)+cm(.rho.-s) (8b)
where s, the beam-separation vector, is by design also the distance between
the reference pinhole and the center of the window. In this case, the
detected intensity contains the broad pinhole diffraction pattern, and the
high-frequency components of the reference beam that "leak" through the
window. As above, the auto-correlation of a(.rho.) contains only three
terms: a delta-function at the zero-frequency position, two
polar-symmetric displaced copies of the mask transmission function, and
the low-level auto-correlation of the mask.
.Fourier.{.vertline.A(r).vertline..sup.2
}.apprxeq.d'.delta.(.rho.)+[d'cm(.rho.-s)+d'cm*(s-.rho.)]+c.sup.2
m(.rho.).degree.m*(-.rho.) (9)
This case allows the investigation of the quality of the pinhole
diffraction that produces the reference wavefront and of the
high-spatial-frequency content of the isolated reference beam. As will be
shown below, it is also a good way to study the characteristics of the
test beam window in situ.
2.4 Measuring Distances in the Fourier-domain
The capacity of this technique to reveal the features of the image-plane
mask in the Fourier-domain analysis warrants a brief discussion of the
relationship between the spatial-frequencies and actual units of
measurement. By considering the simple interference pattern generated by
two spherical waves originating from displaced point-sources in the image
plane, a simple relation is derived.
Considering a given optical system having a numerical aperture, NA, the
cone of light subtending the full angular range will intersect the
detector over an area n.sub.NA pixels in diameter, where the full diameter
of the detector is n.sub.d (>n.sub.NA) pixels. (n.sub.d may be either the
full size of the detector array or an appropriate sub-region that is used
in the Fourier-transform calculation). In the two-wave example, a point
separation of .lambda./(2 NA) generates one wave of path-length
difference, or one fringe across the detected area (.eta..sub.NA pixels).
Therefore, a separation of (n.sub.NA /n.sub.d).lambda./(2 NA) generates
one fringe across the n.sub.d pixels of measurement and would appear in
the Fourier domain as two symmetric delta-function peaks separated by a
distance of two cycles. This result can be cast in a more convenient form.
In the Fourier-transform, the scaling of the pseudo-image is:
##EQU2##
To demonstrate the scaling factor described by Eq. (10), consider the EUV
PS/PDI operating at 13.4-nm wavelength, measuring a system of 0.08 NA. In
this system the full angular range typically subtends 80% of the available
detector width: the ratio n.sub.NA /n.sub.d equals 0.8. This scaling
factor of 0.033 .mu.m per cycle (or equivalently, 0.033 .mu.m per pixel in
the FFT pseudo-image), indicates a Fourier-domain separation of
approximately 30 cycles per micron of real-space distance. Therefore
image-plane windows about 4.5-microns wide appear in the Fourier-transform
as approximately 134cycles across. Features in the Fourier-transform are
resolvable as small as two cycles wide, which corresponds to 0.067 micron
mask features. In practice, however, the finite width of the reference
beam's Fourier-transform reduces this resolution slightly.
3. Experimental Demonstration Images
Several characteristic images from the alignment of the EUV PS/PDI are
diagrammatically depicted in FIG. 4, along with a detailed image of the
central portion of the logarithmically-scaled Fourier-transform magnitude
for each. The optical system under test is a molybdenum/silicon
multilayer-coated Schwarzschild objective operating with 0.07 NA
(numerical aperture) at 13.4-nm wavelength [2]. The transmitted intensity
reveals small defects in the multilayer-coatings that are not important in
this discussion. The PS/PDI mask window is a 4.5 .mu.m wide square.
There are two reference pinholes, located 4.5 .mu.m from the center of the
window, 90-degrees apart.
The situation depicted by the first two pairs of images 100, 110 and 120,
130 in FIG. 4 is represented by Eq. (4), with the strong peak at the
origin of the spatial-frequency spectrum, the two displaced and
polar-symmetric mask transmission functions, and the low-level background
centered about the zero-frequency are apparent. The shape, position, and
orientation of the mask window relative to the test beam focus are clearly
visible.
As alignment proceeds as shown in images 140 and 150 in FIG. 4, the test
beam is nearly centered in the window as shown in image 150. The
Fourier-transform contains the mask transmission function strongly
overlapping a polar-symmetric copy of itself. When the system is brought
into final alignment as shown in images 160 and 170, the reference beam
comes through the reference pinhole and the situation is characterized by
the description of Eq. (7). Three delta functions are visible at final
alignment in image 170, with one at the origin plus a pair of images at
the beam separation positions .+-.s. As before, the window and its
polar-symmetric reflection overlap across the origin, yet now, as
described by Eq. 7(b), additional, fainter copies centered about .+-.s are
visible. Note, however, that the raw-images at each stage do not provide
adequate visual clues for performing the alignment.
FIG. 5 diagrammatically depicts a set of 5 similar pairs of images with
associated pseudo-images. These image were arrived at by altering the
orientation of the beamsplitter grating.
FIG. 6 represents the situation described in Section 2.3 in which only the
reference beam is present. Images of the logarithmically-scaled magnitude
of the Fourier-transform for two different mask test windows are
diagrammatically depicted, each window having two adjacent reference
pinholes. A vertical and a horizontal orientation pseudo-image are shown
for each mask window. The orientation follows the beam-separation
direction, which is set by the orientation of the grating beamsplitter
within the interferometer. Easily visible in the image 280 is the shape of
the mask window (as seen in the identical pair of displaced images 282a,
282b), the delta function peak at the origin 284, the low-level
auto-correlation of the mask transmission function 286, and even the
positions of the pinhole 288 that is not at the center of the reference
beam focus. A representation of a scanning electron-microscope (SEM) image
300 is shown below the corresponding pseudo-images 280, 290 of the same
window. Clearly visible in image 300 are the corresponding upper left
corner mask defect (which corresponds to the same feature seen in the
displaced image 282b), and the pinhole location 304 that is not at the
center of the reference beam focus.
Another set of logarithmically-scaled pseudo-images 310, 320 for a
different PS/PDI mask is displayed to the right of the previous images.
Again a corresponding representation of a SEM image 330 is shown below the
pseudo-images. Small features in the mask can be seen again in both the
pseudo-images and the SEM image.
It should be noted that errors made in performing the subtraction of
offset-signals, or background images can cause spurious artifacts to
appear in the central frequency portions of the Fourier-transformed image.
These spurious features show up as non-zero "stripes" along the x- or
y-axes and are visible in many of the pseudo-images of FIG. 4 through FIG.
6. Offset calibration is often necessary for images recorded with a CCD,
or similar detector, so that the intensity is accurately described. With
proper calibration, the detector provides a high-quality Fourier-transform
"image". The presence of these spurious artifacts has no appreciable
effect on the results, and poses little more than a distraction.
The correspondence between the pseudo-images and the representation of the
SEM images illustrates the power of using pseudo-images as a "microscope".
The PS/PDI can be used in this manner to create Fourier-transform images
of the field in the image plane with no additional components and no
additional cost.
This important property enables this technique to serve a second role.
Since the properties of the field at the image plane can be investigated,
through holographic reconstruction, the technique can be used to probe
other optical performance properties of a system under test, or as a way
to identify patterned features designed as alignment aids.
4. Alignment by Global Image Contrast
When the PS/PDI is nearly aligned, and the interference of the test and
reference beams is visible, fine adjustment of the components can be
performed to optimize the appearance of the fringes across the measurement
domain. The merit function of primary interest is the interference fringe
contrast; maximizing the fringe contrast directly improves the
signal-to-noise ratio in the measurement [7] and will yield the highest
reference-wave quality.
The calculations required by the Fourier-transform alignment method lend
themselves to the definition of a rapidly-calculable global fringe
contrast parameter, .GAMMA., that can be used in alignment. As used
herein, .GAMMA. is defined as the ratio of the power in the reference beam
to the power of the test beam, determined by investigation of the
Fourier-transform of the detected intensity. During alignment, it is not
necessary to calculate .GAMMA. with accuracy or high precision, as long as
a consistent method of calculation is followed: most cases of interest the
position of maximum contrast will coincide with the maximum .GAMMA..
This method is applicable to any interferometric measurement that requires
the optimization of fringe contrast in the presence of a spatial
carrier-frequency. One fundamental prerequisite for the application of
this method is the separability of the first-order peak from the
zeroth-order in the spatial-frequency domain. This property, which is a
standard requirement of the Fourier-transform method of interferogram
analysis [8, 9], is guaranteed in the PS/PDI because of the necessary
separation of the test and reference beams in the image plane. In other
circumstances, the addition of a spatial carrier frequency may be
required, for which the magnitude of the spatial carrier frequency will
depend on the quality of the optical system under test.
To describe the application of this method, we begin with an arbitrary
interferogram where the measured intensity I(r) is represented as:
I(r)=A(r)+B(r) cos [.phi.(r)-k.sub.0.multidot.r], with A,B,.phi. real
numbers, (11)
where r is a coordinate in the plane of measurement, and k.sub.0 is the
spatial carrier-frequency. The local fringe contrast is defined as the
ratio B(r)/A(r), bounded on [0, 1]. Following the description typically
used in the Fourier-transform method of interferogram analysis, it is
useful to employ the simplification that A and B contain only
low-spatial-frequency components. To facilitate the Fourier-domain
representation of the interferograrn, the cosine is separated as follows:
I(r)-A(r)+C(r)e.sup.ik.sup..sub.0 .sup..multidot.r+
C*(r)e.sup.-ik.sup..sub.0 .sup..multidot.r, (12)
where
C(r).tbd.1/2B(r)e.sup.i.phi.(r). (13)
By inspection, the Fourier-transform of the interferogram may be written
i(k)=a(k)+c(k-k.sub.0)+c*(k+k.sub.0), (14)
where A and a, and C and c are Fourier-transform pairs. By assumption, with
primarily low-spatial-frequency content in a and c, they are both strongly
peaked at zero-frequency. The Fourier-transform in Eq. (14) thus contains
three distinct peaks: one at zero-frequency, and two displaced by the
spatial carrier-frequency, located at +k.sub.0 and -k.sub.0.
The definition of the global parameter .GAMMA., based on
single-interferogram analysis, serves in the assessment of the
interferometer's instantaneous alignment. Here we define .GAMMA. as
.GAMMA..tbd.{overscore (B.sup.2 +L )}/A.sup.2 +L . (15)
where .GAMMA. is defined by the root-mean-square magnitudes of B and A
across a sub-domain of the interferogram measurement. It is so defined
because these quantities are readily calculable from the FFT of the whole
interferogram, or from an appropriate sub-domain of the interferogram.
With s as the area of the sub-domain, A.sup.2 +L and B.sup.2 +L are
defined in the spatial domain as
##EQU3##
where the substitution of Eq. (13) into the definition has been made in Eq.
(16b).
By Parseval's Theorem, the total energy content of the spatial domain and
the spatial-frequency domain are equivalent [6]. Hence,
##EQU4##
Since by assumption both a(k) and c(k) are strongly localized about the
central-frequency, the full integrals of Eqs. (17a) and (17b) may be
approximated by the integral over a small region of radius .kappa.,
centered in the spatial-frequency domain.
##EQU5##
In order to match the form of c(k) in Eq. (14), the integration of
.vertline.c(k).vertline..sup.2 in Eq. (18b) has been shifted by the
spatial carrier-frequency. The separation of the three terms in Eq. (14)
allows one final substitution.
##EQU6##
Thus, A.sup.2 +L and B.sup.2 +L are calculated from separate regions of
the same Fourier-transform. Using Eqs. (19a) and (19b) yields a complete
expression for:
##EQU7##
When the spatial carrier-frequency k.sub.0 is not known in advance, it is
easily determined by locating one of the two symmetric points of peak
magnitude in the FFT, outside of an excluded region that contains the
central frequency. It is not necessary to determine k.sub.0 accurately
when the integration radius .kappa. is several cycles in magnitude.
However, when k.sub.0 is known accurately, a more simple approximation for
.GAMMA. can be used, based on the values of i(k) at two points.
##EQU8##
Similar to the Fourier-transform method of interferogram analysis, the
zeroth- and first-order peaks have been isolated from the rest of the
spectrum. The quantity of interest here, however, is the energy content
within a spatial-frequency radius .kappa..
Eqs. (20) and (21) are easily implemented on a computer using the standard
mathematical Fast Fourier-transform. Depending on the combined
characteristics of the illuminating beam and the test optical system, the
radius, .kappa., must be chosen large enough to encircle most of the
zeroth-or first-order components in the spatial-frequency domain, yet
small enough to avoid overlap.
Different values of .kappa. may be chosen for the two integrations, as
appropriate. Typically, these radii must not be larger than half of the
"distance" between the first-order peak and the central frequency. A
radius of ten cycles was chosen for these EUV interferometry experiments
in which more than forty fringes are typically present across the
measurement NA. In practice, evaluation of the alignment position that
produces the maximum value of .GAMMA. is not sensitive to the definition
of .kappa.. Furthermore, the integration regions need not be circular.
To increase the calculation speed, a sub-domain of the interferogram (such
as a central portion, a narrow ribbon of data, or even a single column
through the center) may be used in the contrast calculations.
5. Example IDL Program
One program that has been used for performing and displaying the
pseudo-images is shown below. The program is written in IDL (Interactive
Data Language, by Research Systems Inc.).
;------------------------------------------
; IDL Procedure -- showimage.pro
;------------------------------------------
pro showimage, name, p=p
if not defined(name) then name =`.about.frnguser/image.spe`
s = 180
n = !d.n_colors
rr = [lingen(255, n-2), 230]
gg = [lingen(255, n-2), 0]
bb = gg
tvlct, rr, gg, bb
a = kload(name)
sz = width(a)
q = alog(abs(shift(fft(a,1), sz/2, sz/2)))>(-3)
q2 = congrid(nest(q, s), 360, 360) < 15
wset, 0
p = bytscl(q2, top=n-3)
tv, p
end
During alignment, an image is recorded and saved to the disk as "image.spe"
in this original application of the alignment method. This file is
overwritten many times during the alignment process. From the IDL
command-line interface, the user runs the program. The scaled
Fast-Fourier-Transformn (FFT) pseudo-image appears in a graphics window,
revealing the intermediate alignment position. Those skilled in the art
will appreciate that, instead of saving the images to disk, the images
could be viewed in real time without any effect on the results. Saving of
images to disk or viewing images in real time does not form a part of the
invention. In the event that real time viewing of images is desirable,
various interfaces and analytical tools can be used, including the
graphical user interface for image acquisition and processing described in
my co-pending application Ser. No. 09/181,036 filed on Oct. 27, 1998 and
incorporated herein by reference as background information.
6. Additional Applications
In addition to the uses previously described, the method may be used for
additional applications which include the alignment of image-plane
apertures in general optical systems, the rapid identification of
patterned image-plane alignment marks, and the probing of important
image-plane characteristics of an optical system.
7. Summary
The Fourier-transform alignment method has proven itself to be an
invaluable tool in the rapid alignment of any interferometer, of which an
EUV PS/PDI is but one example. The methods are applicable to a variety of
circumstances where the alignment of components in the image plane of a
coherently illuminated optical system is required. Using a relatively fast
microprocessor, the logarithmically-scaled magnitude of the Fast
Fourier-transform of the recorded intensity may be displayed alongside of
the raw image in near real-time. This provides a powerful and convenient
mode of alignment feedback.
In addition, this method provides a high-resolution pseudo-image of the
image-plane field and optical components, that may be employed as a
"microscope" for the optics and mask system.
Additionally a global fringe contrast parameter may be calculated from the
Fourier-transform data to judge the optimum fine-alignment of the PS/PDI
or any interferometer in which the Fourier-transform methods of
single-interferogram analysis are applicable.
Those skilled in the art will appreciate that the method of the invention
would normally be practiced with the assistance of any conventional
computer system under processor control. Additionally, it will be
understood that any operable software or code for implementing the present
invention on such computer system can be easily developed using
conventional programming techniques.
Accordingly, it will be seen that the current invention, Fourier-transform
and global contrast interferometer alignment method, can be implemented
with numerous variations obvious to those skilled in the art. Although the
description above contains many specificities, these should not be
construed as limiting the scope of the invention, but as merely providing
illustrations of some of the presently preferred embodiments of this
invention. Thus the scope of this invention should be determined by the
appended claims and their legal equivalents.
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